What is the easiest way to animate beams of light in blender?
If you just need a single beam you could use a simple curve object. Make sure to set the spline type to poly if you want all the corners to be sharp and jagged.

Then add a little bevel for thickness (this can be controlled per control point with the radius value), and then add a white emission material. 



After rendering, you'll need to add some blur and glare effects in the compositor. Should get you closer to what you are looking for.




Bonus Round

If you want to animate the light beam, you need to add hooks to your curve control points. Go into edit mode on the curve select one control point, then type CTRL H and select Add Hook to New object. You'll see a small empty add to the position of the control point. Do this for all the control points. Then back into Object mode, you can now animate the empties to move the curve. 

I found it easier to use Proportional editing mode set to Random with a large influence to move one empty and then have the other ones affected slightly as well. I did this on each frame for 5- frames then duplicated them in the Dope sheet a number of times.